Topological String, Matrix Integral, and Singularity Theory

TL;DR

This paper explores the deep connections between topological string theory, matrix integrals, and singularity theory, revealing universal correspondences and analyzing genus expansions and phase transitions.

Contribution

It establishes a link between topological string partition functions and solutions to dispersionless KP systems, and clarifies the role of topological strings in singularity theory.

Findings

Genus zero free energy described by special solutions of dispersionless KP

Universal correspondence between KP hierarchy variables and singularity deformations

Insights into topological matter behavior and gravitational phase transitions

Abstract

We study the relation between topological string theory and singularity theory using the partition function of $A_{N-1}$ topological string defined by matrix integral of Kontsevich type. Genus expansion of the free energy is considered, and the genus $g=0$ contribution is shown to be described by a special solution of $N$-reduced dispersionless KP system. We show a universal correspondences between the time variables of dispersionless KP hierarchy and the flat coordinates associated with versal deformations of simple singularities of type $A$. We also study the behavior of topological matter theory on the sphere in a topological gravity background, to clarify the role of the topological string in the singularity theory. Finally we make some comment on gravitational phase transition.